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A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids
Jun Hu, Shangyou Zhang
TL;DR
Stable mixed finite elements with symmetric H(div)-Pk stresses on tetrahedral grids are difficult to construct, and a suitable Pk stress space had remained an open problem. The paper constructs a family for k≥4 using local edge-bubble enrichment and proves stability. The discrete problems are uniquely solvable, and P4 and P5 numerical tests report optimal-order convergence.
Problem
Constructing stable mixed finite elements for linear elasticity is challenging because the stress tensor must be symmetric, and a suitable H(div)-Pk stress space on tetrahedra was a long-standing open problem.
Source: §1 · p. 1Method
Results
The discrete problem has a unique solution with uniform K-ellipticity and a discrete B-B condition, while P4 and P5 tests report optimal-order convergence.
Source: §3 · p. 8, §4 · p. 10Takeaways & Limitations
The construction provides stable symmetric H(div)-Pk stress spaces for tetrahedral-grid linear elasticity for every k≥4.
Source: §1 · p. 1The analysis assumes the compliance tensor is bounded and uniformly symmetric positive definite.
Source: §2 · p. 2
Abstract
from arXiv · showhide
A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric $H(\d)$-$P_k$ polynomial tensors and the displacement is approximated by $C^{-1}$-$P_{k-1}$ polynomial vectors, for all $k\ge 4$. Numerical tests are provided.
1. Introduction
The paper addresses the challenge of constructing stable mixed finite elements with symmetric H(div) stress spaces, including the long-standing question of a suitable Pk stress space on tetrahedra. It proposes such spaces for k≥4 and develops a stability proof based on characterizing the divergence of the local stress space.
1. Introduction
- For k≥4, the paper constructs an H(div)-Pk stress space, improving on earlier two-order-suboptimal methods, with symmetric stresses enriched by local H(div) edge bubbles.The displacement uses piecewise Pk−1 polynomials; the paper proves stability and optimal-order error estimates and reports P4 and P5 tests.Source: §1 · p. 1
- The central difficulty is designing stable mixed finite elements while enforcing symmetry of the stress tensor.The Hellinger–Reissner formulation places stress in H(div, Ω, S) and displacement in L2(Ω, R3).Source: §1 · p. 1
- The stability proof characterizes the divergence of the local stress space as exactly the Pk displacement subspace orthogonal to local rigid motions.Source: §1 · p. 1
2. The family of finite elements
The method pairs symmetric H(div) stress fields with square-integrable displacements, enriching a continuous polynomial stress space with local edge bubbles. The construction uses bubbles to match displacement divergences locally and describes a basis organization across shared edges and faces.
2. The family of finite elements
- The mixed formulation places symmetric stress in H(div, Ω, S) and displacement in L2(Ω, R3), with the material compliance tensor uniformly bounded and symmetric positive definite.Source: §2 · p. 2
- The stress space is a H(div) bubble enrichment of the H1 stress space, using locally defined edge bubbles whose normal components vanish on element faces.This allows divergence matching on each tetrahedron independently of neighboring elements.Source: §2 · pp. 3–4
- For basis construction, edge bubbles are distributed among tetrahedra sharing an edge and face bubbles between tetrahedra sharing a face; eliminating bubbles linearly dependent on H1-Pk basis functions is not discussed.Source: §2 · p. 3
- The six symmetric tensors Tij form a basis of the symmetric tensor space S.Source: §2 · pp. 4–5
- The strong divergence-free space Zh consists of discrete stresses whose divergence pairs to zero with every discrete displacement.Source: §2 · p. 5
3. Stability and convergence
The section establishes stability of the discrete mixed problem for k ≥ 4, then uses stability and approximation to obtain optimal-order convergence. Its proof constructs divergence-controlled stresses and verifies the discrete B-B condition.
Stability and convergence
- The discrete B-B condition is obtained by first matching divergence moments with a continuous-subspace stress, then correcting the residual with another stress.The first stress has an H(div) bound, and the correction exactly matches the remaining divergence.Source: §3 · pp. 6, 8
- A flux-preserving interpolation into the continuous finite element subspace is built from nodal values, face-triangle averages, and local L2 projections.The construction gives partial divergence matching against polynomials in Pk−3.Source: §3 · pp. 6–7
- For each element, edge-bubble stresses span the divergence space orthogonal to local rigid motions, enabling a bounded local right inverse of divergence.The construction is elementwise and then extends by affine mapping and scaling.Source: §3 · pp. 7–8
- For k ≥ 4, the discrete problem satisfies uniform K-ellipticity and the discrete B-B condition, giving a unique finite element solution.The resulting stability and standard approximation properties yield optimal-order convergence.Source: §3 · pp. 6, 8
- The theorem gives a quasioptimal error estimate for the stable elements, and the interpolation and projection choices yield the stated convergence result.The result applies for k ≥ 4.Source: §3 · p. 9
4. Numerical tests
The P4 and P5 tests report optimal-order convergence. In the P5 test, the stress is exact and displacement convergence is optimal up to computer accuracy.
4. Numerical tests
- The tests use a manufactured solution on the unit cube with homogeneous displacement boundary data and Lamé constants μ = 1/2 and λ = 1; stress and load are defined from that solution.Source: §4 · p. 9
- The initial grid is refined uniformly into half-sized grids, with the level 2 refinement shown alongside the initial grid.Source: §4 · p. 9, Figure 4.1 · p. 9
- P4 achieves optimal-order convergence, while P5 yields exact stress and optimal displacement convergence up to computer accuracy.The P5 exact stress follows because the degree-5 true stress lies in the stress finite element space.Source: Table 4.1 · p. 10, §4 · p. 10, Table 4.2 · p. 10